An improved SPH model for multiphase flows with large density ratios

This paper presents a new smoothed particle hydrodynamics (SPH) model for simulating multiphase fluid flows with large density ratios. The new SPH model consists of an improved discretization scheme, an enhanced multiphase interface treatment algorithm, and a coupled dynamic boundary treatment technique. The presented SPH discretization scheme is developed from Taylor series analysis with kernel normalization and kernel gradient correction and is then used to discretize the Navier‐Stokes equation to obtain improved SPH equations of motion for multiphase fluid flows. The multiphase interface treatment algorithm involves treating neighboring particles from different phases as virtual particles with specially updated density to maintain pressure consistency and a repulsive interface force between neighboring interface particles into the pressure gradient to keep sharp interface. The coupled dynamic boundary treatment technique includes a soft repulsive force between approaching fluid and solid particles while the information of virtual particles are approximated using the improved SPH discretization scheme. The presented SPH model is applied to 3 typical multiphase flow problems including dam breaking, Rayleigh‐Taylor instability, and air bubble rising in water. It is demonstrated that inherent multiphase flow physics can be well captured while the dynamic evolution of the complex multiphase interfaces is sharp with consistent pressure across the interfaces.     

An SPH multi‐fluid model based on quasi buoyancy for interface stabilization up to high density ratios and realistic wave speed ratios

We introduce a smoothed particle hydrodynamics (SPH) concept for the stabilization of the interface between 2 fluids. It is demonstrated that the change in the pressure gradient across the interface leads to a force imbalance. This force imbalance is attributed to the particle approximation implicit to SPH. To stabilize the interface, a pressure gradient correction is proposed. In this approach, the multi‐fluid pressure gradients are related to the (gravitational and fluid) accelerations. This leads to a quasi‐buoyancy correction for hydrostatic (stratified) flows, which is extended to nonhydrostatic flows. The result is a simple density correction that involves no parameters or coefficients. This correction is included as an extra term in the SPH momentum equation. The new concept for the stabilization of the interface is explored in 5 case studies and compared with other multi‐fluid models. The first case is the stagnant flow in a tank: The interface remains stable up to density ratios of 1:1000 (typical for water and air), in combination with artificial wave speed ratios up to 1:4. The second and third cases are the Rayleigh‐Taylor instability and the rising bubble, where a reasonable agreement between SPH and level‐set models is achieved. The fourth case is an air flow across a water surface up to density ratios of 1:100, artificial wave speed ratios of 1:4, and high air velocities. The fifth case is about the propagation of internal gravity waves up to density ratios of 1:100 and artificial wave speed ratios of 1:4. It is demonstrated that the quasi‐buoyancy model may be used to stabilize the interface between 2 fluids up to high density ratios, with real (low) viscosities and more realistic wave speed ratios than achieved by other weakly compressible SPH multi‐fluid models. Real wave speed ratios can be achieved as long as the fluid velocities are not very high. Although the wave speeds may be artificial in many cases, correct and realistic wave speed ratios are essential in the modelling of heat transfer between 2 fluids (eg, in engineering applications such as gas turbines).     

Development of a hybrid particle‐mesh method for two‐phase flow simulations

A hybrid particle‐mesh method was developed for efficient and accurate simulations of two‐phase flows. In this method, the main component of the flow is solved using the constrained interpolated profile/multi‐moment finite volumemethod; the two‐phase interface is rendered using the finite volume particle (FVP) method. The effect of surface tension is evaluated using the continuum surface force model. Numerical particles in the FVP method are distributed only on the surface of the liquid in simulating the interface between liquid and gas; these particles are used to determine the density of each mesh grid. An artificial term was also introduced to mitigate particle clustering in the direction of maximum compression and sparse discretization errors in the stretched direction. This enables accurate interface tracking without diminishing numerical efficiency. Two benchmark simulations are used to demonstrate the validity of the method developed and its numerical stability. Copyright © 2016 John Wiley & Sons, Ltd.     

Modeling surface tension of a two‐dimensional droplet using smoothed particle hydrodynamics

Surface tension plays a significant role at the dynamic interface of free‐surface flows especially at the microscale in capillary‐dominated flows. A model for accurately predicting the formation of two‐dimensional viscous droplets in vacuum or gas of negligible density and viscosity resulting from axisymmetric oscillation due to surface tension is solved using smoothed particle hydrodynamics composed of the Navier‐Stokes system and appropriate interfacial conditions for the free‐surface boundaries. The evolution of the droplet and its free‐surface interface is tracked over time to investigate the effects of surface tension forces implemented using a modified continuous surface force method and is compared with those performed using interparticle interaction force. The dynamic viscous fluid and surface tension interactions are investigated via a controlled curvature model and test cases of nonsteady oscillating droplets; attention is focused here on droplet oscillation that is released from an initial static deformation. Accuracy of the results is attested by demonstrating that (i) the curvature of the droplet that is controlled; (ii) uniform distribution of fluid particles; (iii) clean asymmetric forces acting on the free surface; and (iv) nonsteady oscillating droplets compare well with analytical and published experiment findings. The advantage of the proposed continuous surface force method only requires the use of physical properties of the fluid, whereas the interparticle interaction force method is restricted by the requirement of tuning parameters.     

An SPH pressure correction algorithm for multiphase flows with large density ratio

In this paper, we propose an interfacial pressure correction algorithm for smoothed particle hydrodynamics (SPH) simulation of multiphase flows with large density ratios. This correction term is based on the assumption of small deformation of the interface, and derived from perturbation expansion analysis. It is also proven to be applicable in cases with complex interfaces. This correction algorithm helps to overcome the discontinuities of the pressure gradient over the interfaces, which may cause unphysical gap between different phases. This proposed correction algorithm is implemented on a recent multiphase SPH model, which is based on the assumption of pressure continuity over the interfaces. The coupled dynamic solid boundary treatment is used to simulate solid walls; and a cut‐off pressure is applied to avoid negative particle pressure, which may cause computational instabilities in SPH. Three numerical examples of air–water flows, including sloshing, dam breaking, and water entry, are presented and compared with experimental data, indicating the robustness of our pressure correction algorithm in multiphase simulations with large density ratios. Copyright © 2015 John Wiley & Sons, Ltd.     

Validation of the S‐CLSVOF method with the density‐scaled balanced continuum surface force model in multiphase systems coupled with thermocapillary flows

Three numerical methods, namely, volume of fluid (VOF), simple coupled volume of fluid with level set (S‐CLSVOF), and S‐CLSVOF with the density‐scaled balanced continuum surface force (CSF) model, have been incorporated into OpenFOAM source code and were validated for their accuracy for three cases: (i) an isothermal static case, (ii) isothermal dynamic cases, and (iii) non‐isothermal dynamic cases with thermocapillary flow including dynamic interface deformation. Results have shown that the S‐CLSVOF method gives accurate results in the test cases with mild computation conditions, and the S‐CLSVOF technique with the density‐scaled balanced CSF model leads to accurate results in the cases of large interface deformations and large density and viscosity ratios. These show that these high accuracy methods would be appropriate to obtain accurate predictions in multiphase flow systems with thermocapillary flows. Copyright © 2016 John Wiley & Sons, Ltd.     

An improved three-dimensional model for interface pressure calculations in free-surface flows

A three-dimensional method for the calculation of interface pressure in the computational modeling of free surfaces and interfaces is developed. The methodology is based on the calculation of the pressure force at the interfacial cell faces and is mainly designed for volume of fluid (VOF) interface capturing approach. The pressure forces at the interfacial cell faces are calculated according to the pressure imposed by each fluid on the portion of the cell face that is occupied by that fluid. Special formulations for the pressure in the interfacial cells are derived for different orientations of an interface. The present method, referred to as pressure calculation based on the interface location (PCIL), is applied to both static and dynamic cases. First, a three-dimensional motionless drop of liquid in an initially stagnant fluid with no gravity force is simulated as the static case and then two different small air bubbles in water are simulated as dynamic cases. A two-fluid, piecewise linear interface calculation VOF method is used for numerical simulation of the interfacial flow. For the static case, both the continuum surface force (CSF) and the continuum surface stress (CSS) methods are used for surface tension calculations. A wide range of Ohnesorge numbers and density and viscosity ratios of the two fluids are tested. It is shown that the presence of spurious currents (artificial velocities present in case of considerable capillary forces) is mainly due to the inaccurate calculation of pressure forces in the interfacial computational cells. The PCIL model reduces the spurious currents up to more than two orders of magnitude for the cases tested.

Also for the dynamic bubble rise case, it is shown that using the numerical solver employed here, without PCIL, the magnitude of spurious currents is so high that it is not possible to simulate this type of surface tension dominated flows, while using PCIL, we are able to simulate bubble rise and obtain results in close agreement with the experimental data.     

一种基于黎曼解处理大密度比多相流SPH的改进算法

多相流界面存在密度、黏性等物理场间断,直接采用传统光滑粒子水动力学(smoothedparticle hydrodynamics,SPH)方法进行数值模拟,界面附近的压力和速度存在震荡.一套基于黎曼解能够处理大密度比的多相流SPH计算模型被提出,该模型利用黎曼解在处理接触间断问题方面的优势,将黎曼解引入到SPH多相流计算模型中,为了能够准确求解多相流体物理黏性、减小黎曼耗散,对黎曼形式的SPH动量方程进行了改进,又将Adami固壁边界与黎曼单侧问题相结合来施加多相流SPH固壁边界,同时模型中考虑了表面张力对小尺度异相界面的影响,该模型没有添加任何人工黏性、人工耗散和非物理人工处理技术,能够反应多相流真实物理黏性和物理演变状态.采用该模型首先对三种不同粒子间距离散下方形液滴震荡问题进行了数值模拟,验证了该模型在处理异相界面的正确性和模型本身的收敛性;后又通过对Rayleigh--Taylor不稳定、单气泡上浮、双气泡上浮问题进行了模拟计算,结果与文献对比吻合度高,异相界面捕捉清晰,结果表明,本文改进的多相流SPH模型能够稳定、有效的模拟大密度比和黏性比的多相流问题.      

An unstructured-grid numerical model for interfacial multiphase fluids based on multi-moment finite volume formulation and THINC method

We present a practical numerical framework for incompressible interfacial multiphase flows on unstructured grids with arbitrary and hybrid elements. The numerical framework is constructed by combining VPM (volume-average/point-value multi-moment) and UMTHINC (unstructured multi-dimensional tangent of hyperbola interface capturing) schemes. To facilitate accurate and reliable simulations for interfacial multiphase flows on arbitrary and hybrid unstructured grids, we have made the following major new efforts in this work. (1) UMTHINC scheme on prismatic and pyramidal elements to facilitate computations on hybrid arbitrary unstructured grids; (2) Consistent numerical formulation for mass and momentum transports to simulate multiphase flows of large density ratio; (3) Combined FVM-FEM for accurate solution to diffusion equation; (4) Pressure-projection formulation in consistent with the balanced-force model. Integrating all these numerical techniques effectively enhances the accuracy and robustness in interface capturing and numerical solution of multiphase fluid dynamics, which results in a numerical framework of great significance for practical applications. Numerical verifications have been carried out through benchmark tests ranging from surface tension dominant flows of small scale to large scale flows with violently-changing interfaces. Numerical results demonstrate that the present framework is robust with adequate accuracy for simulating multiphase flows in complex geometries.     

An SPH model for free surface flows with moving rigid objects

This paper presents a computational model for free surface flows interacting with moving rigid bodies. The model is based on the SPH method, which is a popular meshfree, Lagrangian particle method and can naturally treat large flow deformation and moving features without any interface/surface capture or tracking algorithm. Fluid particles are used to model the free surface flows which are governed by Navier–Stokes equations, and solid particles are used to model the dynamic movement (translation and rotation) of moving rigid objects. The interaction of the neighboring fluid and solid particles renders the fluid–solid interaction and the non‐slip solid boundary conditions. The SPH method is improved with corrections on the SPH kernel and kernel gradients, enhancement of solid boundary condition, and implementation of Reynolds‐averaged Navier–Stokes turbulence model. Three numerical examples including the water exit of a cylinder, the sinking of a submerged cylinder and the complicated motion of an elliptical cylinder near free surface are provided. The obtained numerical results show good agreement with results from other sources and clearly demonstrate the effectiveness of the presented meshfree particle model in modeling free surface flows with moving objects. Copyright © 2013 John Wiley & Sons, Ltd.     

A hybrid Lagrangian–Eulerian particle‐level set method for numerical simulations of two‐fluid turbulent flows

A coupled Lagrangian interface‐tracking and Eulerian level set (LS) method is developed and implemented for numerical simulations of two‐fluid flows. In this method, the interface is identified based on the locations of notional particles and the geometrical information concerning the interface and fluid properties, such as density and viscosity, are obtained from the LS function. The LS function maintains a signed distance function without an auxiliary equation via the particle‐based Lagrangian re‐initialization technique. To assess the new hybrid method, numerical simulations of several ‘standard interface‐moving’ problems and two‐fluid laminar and turbulent flows are conducted. The numerical results are evaluated by monitoring the mass conservation, the turbulence energy spectral density function and the consistency between Eulerian and Lagrangian components. The results of our analysis indicate that the hybrid particle‐level set method can handle interfaces with complex shape change, and can accurately predict the interface values without any significant (unphysical) mass loss or gain, even in a turbulent flow. The results obtained for isotropic turbulence by the new particle‐level set method are validated by comparison with those obtained by the ‘zero Mach number’, variable‐density method. For the cases with small thermal/mass diffusivity, both methods are found to generate similar results. Analysis of the vorticity and energy equations indicates that the destabilization effect of turbulence and the stability effect of surface tension on the interface motion are strongly dependent on the density and viscosity ratios of the fluids. Copyright © 2007 John Wiley & Sons, Ltd.     

一种考虑界面相物性影响的界面单元法

界面是由复杂的界面相简化而成的,界面破坏实际是界面相材料的破坏。数值计算为了方便,如经典模型和内聚力模型等,都把很薄的界面相作无厚度化处理。导致只能考虑界面的面力,而无法考虑界面相内的应力(平行于界面方向的应力)。使界面失效准则先天性地排除了界面相内部应力的影响,从界面相材料失效机理的角度来看这是不够严谨的。本文将界面相材料等效为一种弹性连续体,由界面本构关系推导得到了一种新的界面单元。该单元具有界面参数易确定、对界面相物性可以进行等效描述等优点。通过商用有限元软件ABAQUS和用户子程序UEL实现了数值分析,并与直接物理模型的数值模拟结果进行对比,证明了本方法的简便及准确性。通过对不同界面相厚度结构的进一步分析,探讨了本文方法的可行范围。     

Issue Information

An integrated finite element method (FEM) is proposed to simulate incompressible two‐phase flows with surface tension effects, and three different surface tension models are applied to the FEM to investigate spurious currents and temporal stability. A Q2Q1 element is adopted to solve the continuity and Navier–Stokes equations and a Q2‐iso‐Q1 to solve the level set equation. The integrated FEM solves pressure and velocity simultaneously in a strongly coupled manner; the level set function is reinitialized by adopting a direct approach using interfacial geometry information instead of solving a conventional hyperbolic‐type equation. In addition, a consistent continuum surface force (consistent CSF) model is utilized by employing the same basis function for both surface tension and pressure variables to damp out spurious currents and to estimate the accurate pressure distribution. The model is further represented as a semi‐implicit manner to improve temporal stability with an increased time step. In order to verify the accuracy and robustness of the code, the present method is applied to a few benchmark problems of the static bubble and rising bubble with large density and viscosity ratios. The Q2Q1‐integrated FEM coupled with the semi‐implicit consistent CSF demonstrates the significantly reduced spurious currents and improved temporal stability. The numerical results are in good qualitative and quantitative agreements with those of the existing studies. Copyright © 2015 John Wiley & Sons, Ltd.     

Volume preserving immersed boundary methods for two‐phase fluid flows

In this article, we propose a simple area‐preserving correction scheme for two‐phase immiscible incompressible flows with an immersed boundary method (IBM). The IBM was originally developed to model blood flow in the heart and has been widely applied to biofluid dynamics problems with complex geometries and immersed elastic membranes. The main idea of the IBM is to use a regular Eulerian computational grid for the fluid mechanics along with a Lagrangian representation of the immersed boundary. Using the discrete Dirac delta function and the indicator function, we can include the surface tension force, variable viscosity and mass density, and gravitational force effects. The principal advantage of the IBM for two‐phase fluid flows is its inherent accuracy due in part to its ability to use a large number of interfacial marker points on the interface. However, because the interface between two fluids is moved in a discrete manner, this can result in a lack of volume conservation. The idea of an area preserving correction scheme is to correct the interface location normally to the interface so that the area remains constant. Various numerical experiments are presented to illustrate the efficiency and accuracy of the proposed conservative IBM for two‐phase fluid flows. Copyright © 2011 John Wiley & Sons, Ltd.     

Simulating surface tension with smoothed particle hydrodynamics

A method for simulating two‐phase flows including surface tension is presented. The approach is based upon smoothed particle hydrodynamics (SPH). The fully Lagrangian nature of SPH maintains sharp fluid–fluid interfaces without employing high‐order advection schemes or explicit interface reconstruction. Several possible implementations of surface tension force are suggested and compared. The numerical stability of the method is investigated and optimal choices for numerical parameters are identified. Comparisons with a grid‐based volume of fluid method for two‐dimensional flows are excellent. The methods presented here apply to problems involving interfaces of arbitrary shape undergoing fragmentation and coalescence within a two‐phase system and readily extend to three‐dimensional problems. Boundary conditions at a solid surface, high viscosity and density ratios, and the simulation of free‐surface flows are not addressed. Copyright © 2000 John Wiley & Sons, Ltd.     

Structure of inertial-admixture accumulation zones in a tornado-like flow

The motion of a dispersed inertial admixture in a steady-state axisymmetric 3D viscous incompressible flow formed by a semi-infinite vortex filament interacting with an orthogonally located substrate surface is considered. The carrier-phase parameters are found from the numerical solution of the Navier-Stokes equations under the assumption of flow self-similarity of a known type [1]. Different phase force interaction schemes corresponding to different ratios of the phase densities are considered. For calculating the dispersed-phase continuum parameters, a full Lagrangian approach is used, which makes it possible to calculate the dispersed-phase concentration in particle accumulation zones and regions of intersecting particle trajectories. On the basis of parametric calculations, it is found that in the case of heavy particles (whose density is greater than that of the carrier phase) a “cup-shaped” particle accumulation surface visualizing a high-vorticity region is formed. The dependence of this surface shape on the governing parameters is investigated. It is shown that for different phase density ratios the dispersed-phase concentration fields are qualitatively different.     

SPH-based numerical treatment of the interfacial interaction of flow with porous media

In this paper, the macroscopic equations of mass and momentum are developed and discretized based on the smoothed particle hydrodynamics (SPH) formulation for the interaction at an interface of flow with porous media. The theoretical background of flow through porous media is investigated to highlight the key constraints that should be satisfied, particularly at the interface between the porous media flow and the overlying free flow. The study aims to investigate the derivation of the porous flow equations, computation of the porosity, and treatment of the interfacial boundary layer. It addresses weak assumptions that are commonly adopted for interfacial flow simulation in particle-based methods. As support to the theoretical analysis, a two-dimensional weakly compressible SPH model is developed based on the proposed interfacial treatment. The equations in this model are written in terms of the intrinsic averages and in the Lagrangian form. The effect of particle volume change due to the spatial change of porosity is taken into account, and the extra stress terms in the momentum equation are approximated by using Ergun's equation and the subparticle scale model to represent the drag and turbulence effects, respectively. Four benchmark test cases covering a range of flow scenarios are simulated to examine the influence of the porous boundary on the internal, interface, and external flows. The capacity of the modified SPH model to predict velocity distributions and water surface behavior is fully examined with a focus on the flow conditions at the interfacial boundary between the overlying free flow and the underlying porous media.     

An improved diffuse interface method for three-dimensional multiphase flows with complex interface deformation

An improved diffuse interface (DI) method is proposed for accurately capturing complex interface deformation in simulations of three-dimensional (3D) multiphase flows. In original DI methods, the unphysical phenomenon of interface thickening or blurring can affect the accuracy of numerical simulations, especially for flows with large density ratio and high Reynolds number. To remove this drawback, in this article, an interface-compression term is introduced into the Cahn-Hilliard equation to suppress the interface dispersion. The additional term only takes effect in the interface region and works normal to the interface. The difference of the current method from the previous work is that the compression rate can be adjusted synchronously according to the magnitude of local vorticity, which is strongly correlated to the interface dispersion and changes with the computational time and interface position. Numerical validations of the proposed method are implemented by simulating problems of Laplace law, Rayleigh-Taylor instability, bubble rising in a channel, and binary droplet collision. The obtained results agree well with the analytical solutions and published data. The numerical results show that the phenomenon of interface dispersion is suppressed effectively and the tiny interfacial structures in flow field can be captured accurately.     

A cost‐effective curvature calculation approach for interfacial flows on unstructured meshes

We present a simple and cost‐effective curvature calculation approach for simulations of interfacial flows on structured and unstructured grids. The interface is defined using volume fractions, and the interface curvature is obtained as a function of the gradients of volume fractions. The gradient computation is based on a recently proposed gradient recovery method that mimicks the least squares approach without the need to solve a system of equations and is quite easy to implement on arbitrary polygonal meshes. The resulting interface curvature is used in a continuum surface force formulation within the framework of a well‐balanced finite‐volume algorithm to simulate multiphase flows dominated by surface tension. We show that the proposed curvature calculation is at least as accurate as some of the existing approaches on unstructured meshes while being straightforward to implement on any mesh topology. Numerical investigations also show that spurious currents in stationary problems that are dependent on the curvature calculation methodology are also acceptably low using the proposed approach. Studies on capillary waves and rising bubbles in viscous flows lend credence to the ability of the proposed method as an inexpensive, robust, and reasonably accurate approach for curvature calculation and numerical simulation of multiphase flows.